MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtri4 Structured version   Unicode version

Theorem ordtri4 4647
Description: A trichotomy law for ordinals. (Contributed by NM, 1-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordtri4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  ( A  C_  B  /\  -.  A  e.  B ) ) )

Proof of Theorem ordtri4
StepHypRef Expression
1 eqss 3349 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
2 ordtri1 4643 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
32ancoms 441 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
43anbi2d 686 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  C_  B  /\  B  C_  A )  <->  ( A  C_  B  /\  -.  A  e.  B ) ) )
51, 4syl5bb 250 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  ( A  C_  B  /\  -.  A  e.  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727    C_ wss 3306   Ord word 4609
This theorem is referenced by:  carduni  7899  alephfp  8020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-tr 4328  df-eprel 4523  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613
  Copyright terms: Public domain W3C validator